First, observe that the strategies in which agent \(i\)  repeatedly proposes a contract \(\theta ^*\)  as in the proof of thm:main is an SPE: At each proposal and acceptance state, the agent can assume that the value is as if the agent proposes an efficient contract in the next proposal state. As in previous proofs more informally, we invoke the single-deviation principle for SPE (comparee [1]}) stating that to check an SPE, it is sufficient to only consider deviations by one agent at a time. First, observe that there is no reason for the proposer to propose another contract to the other agents, as the upper bound on value demostrated in thm:main continues to hold. The forcing contracts still exist and incentivize all agents to follow \(\pi ^*\) . Finally, rejection of all contracts is again not possible as the proposing agent's best response would not be defined. Hence, the statement continues to hold for a repeatedly proposing agent.

Finally, we show by means of an example that this statement breaks if multiple agents may propose a contract.
